Hi. I’ll walk you through this

example to explore solving a rational equation. Solving a rational equation

involves the following steps. Firstly, find all

restrictions. That means we will find all

the values that make any denominator in the equation 0. These values will be

avoided from being considered as solutions. So let us look at

the equation. The denominators in the given

equation are x and 2x. Look at the numbers negative

3 and positive 3. They can otherwise be written

as negative 3 over 1 and 3 over 1. The denominator for these

numbers then would be 1 in both cases. When you see 1 as

the denominator, you can ignore it. Now we look for values that

make the denominators 0. The denominators x and 2x will

become 0 when x is equal to 0. So the restriction will

be x is equal to 0. The second step is to find the

LCD of all the rational expressions in the

given equation. In order to do that, we look at

these denominators, x and 2x, and determine their LCD. Between x and 2x, we can

quickly recognize that the LCD is 2x. The third step is to multiply

the equation by LCD. The equation we have is 5/x

minus 3 equals 7/2x plus 3. We multiply this equation

by the LCD we found. So let us wrap the left-hand

side as well as the right-hand side, and multiply by the

LCD, which is 2x. We distribute the 2x on to

every term inside the parentheses on both sides, and

that would give us 2x times 5/x minus 2x times 3 equals 2x

times 7/2x plus 2x times 3. Now we will simplify each

term of this equation. In the first term of the

left-hand side, we see x on the top and x in the bottom. They divide each other,

giving 1. Same way in the first term of

the right-hand side, we see 2x on the top and 2x

in the bottom. They, too, divide each

other, giving 1. Now we simplify the equation

that we have, and that would be 2 times 5, which is 10, minus

2x times 3, and that will be 6x equals– we just have 7 in the first term

of the right-hand side– plus the second term

will give 6x For the next step, we take that

resulting equation and solve for x. You must observe that we got the

denominator as 1 for every term in that equation. We will now go ahead

and solve this. In order to do that, we group

x terms to one side and numbers to the other. Let us add 6x to both sides so

we can group the x terms. We are left with 10 on the

left-hand side, and 7 plus 12x on the right-hand side. Now to bring the numbers to the

same side, we subtract 7 from both sides. We will have 3 equals 12x. To isolate x, we divide by 12

because 12x is 12 times x, and we do the opposite to

get rid of that 12. So we divide by 12, and what we

do to one side we do to the other as well. And this would give

x to be 1/4. As step 5, we will compare the

proposed solutions to the list of restrictions and reject any

proposed solution that is in the list of restrictions. The solution we got

is x equals 1/4. And in the list of restrictions,

if you try to look at step 1, we have

restriction to be just 0. Since 1/4 is not equal to 0, we

do not reject the solution. As the last step, we go

ahead and check the validity of the solution. In order to do that, we plug

in the solution in the original equation and

look for truth. We got x equals 1/4. That’s the proposed solution. Let us plug that in the

original equation. The original equation is 5/x

minus 3 equals 7/2x plus 3. We are plugging 1/4

in the place of x. And this is what we get. We take that and try to

simplify it further. You must recall how to divide

when you have fractions in the denominator. So we make the division as

multiplication and flip the following fraction, which will

give us 5 times 4/1 minus 3 equals 7 times 4 divided

by 2 plus 3. Simplifying further, we will get

17 equals 17, which is a true statement. What that tells us is x equals

1/4 is a valid solution for the given equation. I hope this helps. Thank you.